Nonlinear rheology of cellular networks

Viscoelastic mechanics of living cells

In cell mechanotransduction, cells respond to external forces or to perceive mechanical properties of their supporting substrates by remodeling themselves. This ability is endowed by modulating cells' viscoelastic properties, which dominates over various complex cellular processes. The viscoelasticity of living cells, a concept adapted from rheology, exhibits substantially spatial and temporal variability. This review aims not only to discuss the rheological properties of cells but also to clarify the complexity of cellular rheology, emphasizing its dependence on both the size scales and time scales of the measurements. Like typical viscoelastic materials, the storage and loss moduli of cells often exhibit robust power-law rheological characteristics with respect to loading frequency. This intrinsic feature is consistent across cell types and is attributed to internal structures, such as cytoskeleton, cortex, cytoplasm and nucleus, all of which contribute to the complexity of cellular rheology. Moreover, the rheological properties of cells are dynamic and play a crucial role in various cellular and tissue functions. In this review, we focus on elucidating time- and size-dependent aspects of cell rheology, the origins of intrinsic rheological properties and how these properties adapt to cellular functions, with the goal of interpretation of rheology into the language of cell biology.

Junctional force patterning drives both positional order and planar polarity in the auditory epithelia

Tissue function depends on the precise organisation of the constituent cells. In the cochlea, the fidelity of hearing depends on mechanosensory hair cells being consistently surrounded by supporting cells. In addition to this positional order, auditory sensitivity depends crucially on planar cell polarity. This is characterised by the alignment of the orientation of eccentrically placed hair bundles on each hair cell. These two levels of order emerge simultaneously despite the cellular fluxes that occur during cochlear development. However, the link between tissue-scale cellular rearrangements and intrinsic cellular mechanisms remains unknown. By combining experimental and theoretical approaches, we find a precise force patterning underpinning positional order and planar cell polarity. This occurs through the modulation of the levels and phospho-type of the regulatory light chain of non-muscle myosin II at specific cell-cell junctions of the auditory epithelium. We propose that the control of junctional mechanics is vital for the organisation of multi-cell-type epithelia.

Connecting Anomalous Elasticity and Sub-Arrhenius Structural Dynamics in a Cell-Based Model

Understanding the structural dynamics of many-particle glassy systems remains a key challenge in statistical physics. Over the last decade, glassy dynamics has also been reported in biological tissues, but is far from being understood. It was recently shown that vertex models of dense biological tissue exhibit very atypical, sub-Arrhenius dynamics, and here we ask whether such atypical structural dynamics of vertex models are related to unusual elastic properties. It is known that at zero temperature these models have an elasticity controlled by their underconstrained or isostatic nature, but little is known about how their elasticity varies with temperature. To address this question we investigate the 2D Voronoi model and measure the temperature dependence of the intermediate-time plateau shear modulus and the bulk modulus. We find that unlike in conventional glass formers, these moduli increase monotonically with temperature until the system fluidizes. We further show that the structural relaxation time can be quantitatively linked to the plateau shear modulus G_{p}, i.e. G_{p} modulates the typical energy barrier scale for cell rearrangements. This suggests that the anomalous, structural dynamics of the 2D Voronoi model originates in its unusual elastic properties. Based on our results, we hypothesize that underconstrained systems might more generally give rise to a new class of "ultrastrong" glass formers.

Shaping epithelial lumina under pressure

The formation of fluid- or gas-filled lumina surrounded by epithelial cells pervades development and disease. We review the balance between lumen pressure and mechanical forces from the surrounding cells that governs lumen formation. We illustrate the mechanical side of this balance in several examples of increasing complexity, and discuss how recent work is beginning to elucidate how nonlinear and active mechanics and anisotropic biomechanical structures must conspire to overcome the isotropy of pressure to form complex, non-spherical lumina.

Discontinuous Shear Thickening in Biological Tissue Rheology

During embryonic morphogenesis, tissues undergo dramatic deformations in order to form functional organs. Similarly, in adult animals, living cells and tissues are continually subjected to forces and deformations. Therefore, the success of embryonic development and the proper maintenance of physiological functions rely on the ability of cells to withstand mechanical stresses as well as their ability to flow in a collective manner. During these events, mechanical perturbations can originate from active processes at the single-cell level, competing with external stresses exerted by surrounding tissues and organs. However, the study of tissue mechanics has been somewhat limited to either the response to external forces or to intrinsic ones. In this work, we use an active vertex model of a 2D confluent tissue to study the interplay of external deformations that are applied globally to a tissue with internal active stresses that arise locally at the cellular level due to cell motility. We elucidate, in particular, the way in which this interplay between globally external and locally internal active driving determines the emergent mechanical properties of the tissue as a whole. For a tissue in the vicinity of a solid-fluid jamming or unjamming transition, we uncover a host of fascinating rheological phenomena, including yielding, shear thinning, continuous shear thickening, and discontinuous shear thickening. These model predictions provide a framework for understanding the recently observed nonlinear rheological behaviors in vivo.

Single and two-cells shape analysis from energy functionals for three-dimensional vertex models

Vertex models have been extensively used for simulating the evolution of multicellular systems, and have given rise to important global properties concerning their macroscopic rheology or jamming transitions. These models are based on the definition of an energy functional, which fully determines the cellular response and conclusions. While two-dimensional vertex models have been widely employed, three-dimensional models are far more scarce, mainly due to the large amount of configurations that they may adopt and the complex geometrical transitions they undergo. We here investigate the shape of single and two-cells configurations as a function of the energy terms, and we study the dependence of the final shape on the model parameters: namely the exponent of the term penalising cell-cell adhesion and surface contractility. In single cell analysis, we deduce analytically the radius and limit values of the contractility for linear and quadratic surface energy terms, in 2D and 3D. In two-cells systems, symmetrical and asymmetrical, we deduce the evolution of the aspect ratio and the relative radius. While in functionals with linear surface terms yield the same aspect ratio in 2D and 3D, the configurations when using quadratic surface terms are distinct. We relate our results with well-known solutions from capillarity theory, and verify our analytical findings with a three-dimensional vertex model.

Shape-Tension Coupling Produces Nematic Order in an Epithelium Vertex Model

We study the vertex model for epithelial tissue mechanics extended to include coupling between the cell shapes and tensions in cell-cell junctions. This coupling represents an active force which drives the system out of equilibrium and leads to the formation of nematic order interspersed with prominent, long-lived +1 defects. The defects in the nematic ordering are coupled to the shape of the cell tiling, affecting cell areas and coordinations. This intricate interplay between cell shape, size, and coordination provides a possible mechanism by which tissues could spontaneously develop long-range polarity through local mechanical forces without resorting to long-range chemical patterning.

Random Traction Yielding Transition in Epithelial Tissues

We investigate how randomly oriented cell traction forces lead to fluidization in a vertex model of epithelial tissues. We find that the fluidization occurs at a critical value of the traction force magnitude F_{c}. We show that this transition exhibits critical behavior, similar to the yielding transition of sheared amorphous solids. However, we find that it belongs to a different universality class, even though it satisfies the same scaling relations between critical exponents established in the yielding transition of sheared amorphous solids. Our work provides a fluidization mechanism through active force generation that could be relevant in biological tissues.

Constitutive model for the rheology of biological tissue

The rheology of biological tissue is key to processes such as embryo development, wound healing, and cancer metastasis. Vertex models of confluent tissue monolayers have uncovered a spontaneous liquid-solid transition tuned by cell shape; and a shear-induced solidification transition of an initially liquidlike tissue. Alongside this jamming/unjamming behavior, biological tissue also displays an inherent viscoelasticity, with a slow time and rate-dependent mechanics. With this motivation, we combine simulations and continuum theory to examine the rheology of the vertex model in nonlinear shear across a full range of shear rates from quastistatic to fast, elucidating its nonlinear stress-strain curves after the inception of shear of finite rate, and its steady state flow curves of stress as a function of strain rate. We formulate a rheological constitutive model that couples cell shape to flow and captures both the tissue solid-liquid transition and its rich linear and nonlinear rheology.

Tissue flows are tuned by actomyosin-dependent mechanics in developing embryos

Rapid epithelial tissue flows are essential to building and shaping developing embryos. However, the mechanical properties of embryonic epithelial tissues and the factors that control these properties are not well understood. Actomyosin generates contractile tensions and contributes to the mechanical properties of cells and cytoskeletal networks in vitro, but it remains unclear how the levels and patterns of actomyosin activity contribute to embryonic epithelial tissue mechanics in vivo. To dissect the roles of cell-generated tensions in the mechanics of flowing epithelial tissues, we use optogenetic tools to manipulate actomyosin contractility with spatiotemporal precision in the Drosophila germband epithelium, which rapidly flows during body axis elongation. We find that manipulating actomyosin-dependent tensions by either optogenetic activation or deactivation of actomyosin alters the solid-fluid mechanical properties of the germband epithelium, leading to changes in cell rearrangements and tissue-level flows. Optogenetically activating actomyosin leads to increases in the overall level but decreases in the anisotropy of tension in the tissue, whereas optogenetically deactivating actomyosin leads to decreases in both the level and anisotropy of tension compared to in wild-type embryos. We find that optogenetically activating actomyosin results in more solid-like (less fluid-like) tissue properties, which is associated with reduced cell rearrangements and tissue flow compared to in wild-type embryos. Optogenetically deactivating actomyosin also results in more solid-like properties than in wild-type embryos but less solid-like properties compared to optogenetically activating actomyosin. Together, these findings indicate that increasing the overall tension level is associated with more solid-like properties in tissues that are relatively isotropic, whereas high tension anisotropy fluidizes the tissue. Our results reveal that epithelial tissue flows in developing embryos involve the coordinated actomyosin-dependent regulation of the mechanical properties of tissues and the tensions driving them to flow in order to achieve rapid tissue remodeling.

The role of non-affine deformations in the elastic behavior of the cellular vertex model

The vertex model of epithelia describes the apical surface of a tissue as a tiling of polygonal cells, with a mechanical energy governed by deviations in cell shape from preferred, or target, area, A₀, and perimeter, P₀. The model exhibits a rigidity transition driven by geometric incompatibility as tuned by the target shape index, . For with p_*(6) the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and preferred perimeter. As a result, the tissue is in a mechanically soft compatible state, with zero shear and Young's moduli. For p₀ < p_*(6), it is geometrically impossible for any cell to realize the preferred area and perimeter simultaneously, and the tissue is in an incompatible rigid solid state. Using a mean-field approach, we present a complete analytical calculation of the linear elastic moduli of an ordered vertex model. We analyze a relaxation step that includes non-affine deformations, leading to a softer response than previously reported. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape. The bulk modulus exhibits a jump discontinuity at the transition and can be lower in the rigid state than in the fluid-like state. The Poisson's ratio can become negative which lowers the bulk and Young's moduli. Our work provides a unified treatment of linear elasticity for the vertex model and demonstrates that this linear response is protocol-dependent.

Emergence of Spatial Scales and Macroscopic Tissue Dynamics in Active Epithelial Monolayers

Migrating cells in tissues are often known to exhibit collective swirling movements. In this paper, we develop an active vertex model with polarity dynamics based on contact inhibition of locomotion (CIL). We show that under this dynamics, the cells form steady-state vortices in velocity, polarity, and cell stress with length scales that depend on polarity alignment rate (ζ), self-motility (v0), and cell-cell bond tension (λ). When the ratio λ/v0 becomes larger, the tissue reaches a near jamming state because of the inability of the cells to exchange their neighbors, and the length scale associated with tissue kinematics increases. A deeper examination of this jammed state provides insights into the mechanism of sustained swirl formation under CIL rule that is governed by the feedback between cell polarities and deformations. To gain additional understanding of how active forcing governed by CIL dynamics leads to large-scale tissue dynamics, we systematically coarse-grain cell stress, polarity, and motility and show that the tissue remains polar even on larger length scales. Overall, we explore the origin of swirling patterns during collective cell migration and obtain a connection between cell-level dynamics and large-scale cellular flow patterns observed in epithelial monolayers.

Curvature-Induced Cell Rearrangements in Biological Tissues

On a curved surface, epithelial cells can adapt to geometric constraints by tilting and by exchanging their neighbors from apical to basal sides, known as an apico-basal topological transition 1 (AB-T1). The relationship between cell tilt, AB-T1s, and tissue curvature still lacks a unified understanding. Here, we propose a general framework for cell packing in curved environments and explain the formation of AB-T1s from the perspective of strain anisotropy. We find that steep curvature gradients can lead to cell tilting and induce AB-T1s. Alternatively, pressure differences across the epithelial tissue can drive AB-T1s in regions of large curvature anisotropy. The two mechanisms compete to determine the impact of tissue geometry and mechanics on optimized cell rearrangements in three dimensions.

Shear-Driven Solidification and Nonlinear Elasticity in Epithelial Tissues

Biological processes, from morphogenesis to tumor invasion, spontaneously generate shear stresses inside living tissue. The mechanisms that govern the transmission of mechanical forces in epithelia and the collective response of the tissue to bulk shear deformations remain, however, poorly understood. Using a minimal cell-based computational model, we investigate the constitutive relation of confluent tissues under simple shear deformation. We show that an initially undeformed fluidlike tissue acquires finite rigidity above a critical applied strain. This is akin to the shear-driven rigidity observed in other soft matter systems. Interestingly, shear-driven rigidity can be understood by a critical scaling analysis in the vicinity of the second order critical point that governs the liquid-solid transition of the undeformed system. We further show that a solidlike tissue responds linearly only to small strains and but then switches to a nonlinear response at larger stains, with substantial stiffening. Finally, we propose a mean-field formulation for cells under shear that offers a simple physical explanation of shear-driven rigidity and nonlinear response in a tissue.

Active T1 transitions in cellular networks

In amorphous solids as in tissues, neighbor exchanges can relax local stresses and allow the material to flow. In this paper, we use an anisotropic vertex model to study T1 rearrangements in polygonal cellular networks. We consider two different physical realizations of the active anisotropic stresses: (i) anisotropic bond tension and (ii) anisotropic cell stress. Interestingly, the two types of active stress lead to patterns of relative orientation of T1 transitions and cell elongation that are different. Our work suggests that these two realizations of anisotropic active stresses can be observed in vivo. We describe and explain these results through the lens of a continuum description of the tissue as an anisotropic active material. We furthermore discuss the energetics of the dynamic tissue and express the energy balance in terms of internal elastic energy, mechanical work, chemical work and heat. This allows us to define active T1 transitions that can perform mechanical work while consuming chemical energy.